Henry Ernest Dudeney/Puzzles and Curious Problems/60 - Equal Distances/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $60$

Equal Distances
At what time between three and four o'clock is the minute hand the same distance from $\text {VIII}$ as the hour hand is from $\text {XII}$?


Solution

$23 \tfrac 1 {13}$ past $3$.
Dudeney-Puzzles-and-Curious-Problems-60-Solution.png


Proof

Let $T$ be the time in question.

Let $m$ be the number of minutes past $3:00$ that $T$ is.

Let $\theta \degrees$ be the angle made by the minute hand with respect to twelve o'clock at time $T$.

Let $\phi \degrees$ be the angle made by the hour hand with respect to twelve o'clock at time $T$.

Between $3:00$ and $4:00$ we have that $90 < \phi < 120$.

Hence from Condition for Valid Time Indication, we have that:

$12 \paren {\phi - 90} = \theta$

Then we are told that:

$\size {240 - \theta} = \phi$


This can mean that:

$240 - \theta = \phi$

in which case:

\(\text {(1)}: \quad\) \(\ds 240 - \theta\) \(=\) \(\ds \phi\)
\(\text {(2)}: \quad\) \(\ds 12 \paren {\phi - 90}\) \(=\) \(\ds \theta\)
\(\ds \leadsto \ \ \) \(\ds 12 \paren {240 - \theta - 90}\) \(=\) \(\ds \theta\) substituting for $\phi$ in $(2)$ from $(1)$
\(\ds \leadsto \ \ \) \(\ds 13 \theta\) \(=\) \(\ds 12 \times 150\)
\(\ds \leadsto \ \ \) \(\ds \theta\) \(=\) \(\ds \dfrac {12 \times 150} {13}\)
\(\ds \leadsto \ \ \) \(\ds m\) \(=\) \(\ds \dfrac {12 \times 150} {13} \times \dfrac 1 6\) Speed of Minute Hand
\(\ds \) \(=\) \(\ds 23 \tfrac 1 {13}\) calculating


Or it can mean that:

$\theta - 240 = \phi$

in which case:

\(\text {(1)}: \quad\) \(\ds \theta - 240\) \(=\) \(\ds \phi\)
\(\text {(2)}: \quad\) \(\ds 12 \paren {\phi - 90}\) \(=\) \(\ds \theta\)
\(\ds \leadsto \ \ \) \(\ds 12 \paren {\theta - 240 - 90}\) \(=\) \(\ds \theta\) substituting for $\phi$ in $(2)$ from $(1)$
\(\ds \leadsto \ \ \) \(\ds 11 \theta\) \(=\) \(\ds 12 \times 330\)
\(\ds \leadsto \ \ \) \(\ds \theta\) \(=\) \(\ds 360\)
\(\ds \leadsto \ \ \) \(\ds m\) \(=\) \(\ds 60\) Speed of Minute Hand

But in this case $T = 4:00$, which is not (strictly) between $3:00$ and $4:00$.

Hence the result.

$\blacksquare$


Sources

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